Let \(H,K\)  be any two elements of \(\mathcal {K}_{1}\)  contained \(T\) . Then they are
elation groups of \(PG_{2}(3)\)  with center \(C_{i}\)  and axis \(\ell _{i}\) ,
where \(i=1,2\) , respectively. If either \(C_{1}=C_{2}\)  or \(\ell _{1}=\ell _{2}\) , then \(\left\langle H,K\right\rangle \)  is elementary abelian of order 9
by [1]}, Theorem 4.14, but this contradicts \(\left|T\right|\mid 24\) . Therefore, \(C_{1}\ne C_{2}\)  and \(\ell _{1}\ne \ell _{2}\) . Then \(\left\langle H,K\right\rangle \)  fixes the point \(\ell _{1}\cap \ell _{2}\) 
and the line \(C_{1}C_{2}\) . If \(\ell _{1}\cap \ell _{2}\in C_{1}C_{2}\) , then \(\ell _{1}\cap \ell _{2}\in \left\lbrace  C_{1},C_{2}\right\rbrace  \) , as \(\ell _{1} \ne \ell _{2}\) . Suppose that \(\ell _{1}\cap \ell _{2}=\left\lbrace  C_{1}\right\rbrace  \) , then \(K\)  fixes \(C_{1}\)  and there
is \(\gamma \in K\)  such that \(\ell _{1}^{\gamma }\ne \ell _{1}\)  as \(C_{1}\ne C_{2}\) . So \(\left\langle H,H^{\gamma }\right\rangle \leqslant T\) , with \(\left\langle H,H^{\gamma }\right\rangle \)  elementary abelian of order 9
again by [1]}, Theorem 4.14, and we reach a contradiction. The case \(\ell _{1}\cap \ell _{2}=\left\lbrace  C_{2}\right\rbrace \)  is ruled out similarly. Thus, \(\ell _{1}\cap \ell _{2}\notin C_{1}C_{2}\) . Then \(SL_{2}(3)\trianglelefteq \left\langle H,K\right\rangle \leqslant GL_{2}(3)\) , as the stabilizer in \(PSL_{3}(3)\)  of \(\ell _{1}\cap \ell _{2}\)  and of \(C_{1}C_{2}\)  is \(GL_{2}(3)\) .
Therefore, \(T=\left\langle H,K\right\rangle \cong SL_{2}(3)\) , as \(\left\langle H,K\right\rangle \leqslant T\)  and \(\left|T\right|\mid 24\) . This proves
(1).
Assume that \(T\)  contains at least two elements \(L,M\)  of \(\mathcal {K}_{2}\) .
Then \(T_{0}\)  contains \(L,M\) , where \(T_{0}=T\cap PSL_{3}(3)\) . By [3]}, either \(T_{0}\leqslant W:GL_{2}(3)\) , where \(W\)  is an elementary abelian group of order 9
consisting of elations of \(PG_{2}(3)\) , or \(T_{0}\leqslant S_{4}\)  and \(T_{0}\) 
preserves a conic of \(PG_{2}(3)\) . If \(T_{0}\leqslant W:GL_{2}(3)\) ,
then \(T_{0}\cap W=1\) , since the Sylow 3-subgroups of \(T_{0}\)  belong to \(\mathcal {K}_{2}\) . Hence, \(T_{0}\)  is isomorphic to a subgroup of \(GL_{2}(3)\) .
Since \(GL_{2}(3)\)  does not contain subgroups of order 12 with at least two
distinct Sylow 3-subgroups, and since \(\left|T\right|=12\)  or 24, it follows that \(T_{0}\cong SL_{2}(3)\) . Then \(T_{0}\)  fixes an anti-flag of \(PG_{2}(3)\) , as \(Z(T)\cong Z_{2}\)  is an homology of \(PG_{2}(3)\) . However this case is ruled out, since any element of \(\mathcal {K}_{2}\)  fixes just a flag and no more
points or lines of \(PG_{2}(3)\) . Thus, \(T_{0}\leqslant S_{4}\) , \(T_{0}\)  preserves a
conic of \(PG_{2}(3)\)  and hence \(A_{4}\trianglelefteq T_{0}\leqslant S_{4}\) , since
\(\left\langle H,K\right\rangle \leqslant T_{0}\) . Then either \(T_{0}=T\)  and \(T\cong A_{4}\) , \(S_{4}\) , or \(T_{0} \ne T\) , \(T_{0}\cong A_{4}\)  and \(T\cong S_{4}\) , \(A_{4}\times Z_{2}\) , since \(\left|T\right|=12\)  or 24. This proves
(2).
